非可定向曲面的非可定向加厚中纽结的考夫曼括号多项式类似物
An analog of the {K}auffman bracket polynomial for knots in the non-orientable thickening of a non-orientable surface
AI总结：
本文针对非可定向曲面加厚中的伪经典纽结，通过修改交叉点符号与光滑化定义，构造了考夫曼括号多项式的类似物，并证明其是合痕不变量且独立于经典考夫曼括号多项式。
摘要：
我们研究非可定向曲面的非可定向加厚中的伪经典纽结，具体是指在形如（非可定向曲面）× [0, 1] 的非可定向三维流形中保持定向路径的纽结。针对这些纽结，我们提出了考夫曼括号多项式的一个类似物。该多项式的构造与经典版本非常接近，关键区别在于交叉点的符号以及交叉点的正/负光滑化的定义。我们证明该多项式是伪经典纽结的合痕不变量，并证明它独立于可定向加厚曲面中纽结的经典考夫曼括号多项式，该可定向曲面是所考虑的非可定向曲面的可定向二重覆盖。
英文摘要：
We study pseudo-classical knots in the non-orientable thickening of a non-orientable surface, specifically knots that are orientation-preserving paths in a non-orientable $3$-manifold of the form (non-orientable surface) $\times$ $[0, 1]$. For these knots, we propose an analog of the Kauffman bracket polynomial. The construction of this polynomial closely mirrors the classical version, with key differences in the definitions of the sign of a crossing and the positive/negative smoothing of a crossing. We prove that this polynomial is an isotopy invariant of pseudo-classical knots and demonstrate that it is independent of the classical Kauffman bracket polynomial for knots in the thickened orientable surface, which is the orientable double cover of the non-orientable surface under consideration.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
